In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). sin x = 0.8246

검증된 단계별 안내

Identify the equation to solve: \(\sin x = 0.8246\) on the interval \([0, 2\pi)\).

Use the inverse sine function to find the principal solution: \(x = \sin^{-1}(0.8246)\).

Calculate the principal value using a calculator, ensuring the mode is set to radians.

Recall that sine is positive in the first and second quadrants, so find the second solution using \(x = \pi - \sin^{-1}(0.8246)\).

List both solutions within the interval \([0, 2\pi)\) and express them rounded to four decimal places.

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주요 개념

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Solving Trigonometric Equations

Solving trigonometric equations involves finding all angle values within a specified interval that satisfy the given equation. For sine equations, this means identifying angles whose sine value matches the given number, considering the periodic nature of the sine function.

Using the Inverse Sine Function

The inverse sine function (sin⁻¹ or arcsin) is used to find the principal angle whose sine is a given value. Since sine is positive in the first and second quadrants, two solutions typically exist within [0, 2π), which must be calculated and verified.

Interval and Periodicity of Sine Function

The sine function has a period of 2π, meaning its values repeat every 2π radians. When solving on the interval [0, 2π), it is important to find all solutions within one full cycle, including angles in both the first and second quadrants where sine is positive.